
where \(S\)  in a real number larger than 1. Note that \(S\)  is not restricted to be an integer. We downsample the groundtruth images as the source images \(\mathbf {I}_{1}\)  (i.e., the data) by bicubic interpolation, following the traditional experimental setting in the literature of SISR, as the learned weights are to be used for algorithm evaluation in comparison with state of the art. We use the groundtruth images (taken from benchmark training dataset BSD200 [1]}) as target images \(\mathbf {I}_{2}\)  (i.e., the labels). For each pixel \([i,j]\)  in the an image \(\mathbf {I}_{2}\) , the RGB values of its correspondence \((x,y)\)  in \(\mathbf {I}_{1}\)  thus are known in the training process. \(\tilde{G}_{1}[x,y]\) , \(\tilde{R}_{1}[x,y]\)  and \(\tilde{B}_{1}[x,y]\) , as well as \(\Delta G_{1}[x,y]\) , \(\Delta R_{1}[x,y]\)  and \(\Delta B_{1}[x,y]\)  are computed on the pixels in each source image by interpolation function \(f\) . Subsequently, Eq.REF  can be rewritten as linear systems of the weights \(\omega _{(\cdot ,\cdot )}\) . For example,
\(\begin{split}&\left[\begin{array}{cc}\Delta R_{1}[x_1,y_1] & \Delta B_{1}[x_1,y_1]\\\vdots & \vdots \\\Delta R_{1}[x_K,y_K] & \Delta B_{1}[x_K,y_K]\end{array}\right]\left[\begin{array}{c}\omega _{g,r}\\ \omega _{g,b}\end{array}\right] \\& =\left[\begin{array}{c}\tilde{G}_{1}[x_1,y_1]-G_{2}[i_1,j_1]\\\vdots \\\tilde{G}_{1}[x_K,y_K]-G_{2}[i_K,j_K]\end{array}\right]\end{split}\) 
